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A Strong Gram Classification of Non-negative Unit Forms of Dynkin Type A

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Jiménez-González J. A.

Fundamenta Informaticae

2024

tom Vol. 191, nr 1

17--67

An integral quadratic form q is usually identified with a bilinear form b such that its Gram matrix with respect to the canonical basis is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type A_r and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49-82, 2021] and [Fundamenta Informaticae 185(3):221-246, 2022].

Coxeter Invariants for Non-negative Unit Forms of Dynkin Type A

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Jiménez González J. A.

2022

tom Vol. 185, nr 3

221--246

Two integral quadratic unit forms are called strongly Gram congruent if their upper triangular Gram matrices are ℤ-congruent. The paper gives a combinatorial strong Gram invariant for those unit forms that are non-negative of Dynkin type r (for r ≥ 1), within the framework introduced in [Fundamenta Informaticae 184(1):49–82, 2021], and uses it to determine all corresponding Coxeter polynomials and (reduced) Coxeter numbers.